Problem Solving

jk2010 wrote:For all positive integers m, @m=3m when m is odd and @m=1/2 m when m is even. What does @9 x @6 equate to?

A - @81
B - @59
C - @36
D - @27
E - @18

I just want to add one point anshu's solution.
From his solution we have (@9 x @6) = 81. Now the options are in '@' representation. Thus we have to find a number n, for which @n = 81. From the definition of @m, this can be done in two ways.

• 1. For odd n, @n = 3n = 81 => n = 27
  2. For even n, @n = n/2 = 81 => n = 162

Only @27 is in the option, hence it is the correct answer. But @162 is also a correct representation.

Now why I'm doing this apparently redundant addition?
Say if the question was "What does @9 x @8 equate to?
Now, (@9 x @8) = (3*9 x (8/2)) = (27*4) = 108

Again we have to find a number n, for which @n = 108. From the definition of @m, this can be done in two ways.

• 1. For odd n, @n = 3n = 108 => n = 36 ... This is wrong! Because we have assumed n as odd, but we are getting n = 36 for which @n = (36/2) = 18
  2. For even n, @n = n/2 = 108 => n = 216 ... This is correct. Because we have assumed n as even and 216 is even, for which @n = (216/2) = 108

Hope this helps.